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College Park, ND 2013 PROCEEDINGS of the NPA
The Speed of Light in 3-dimensional Euclidean Space
Nina Sotina
3250 Coney Island Ave., Unit E2, Brooklyn, NY, 11235
e-mail:
Nadia Lvov
Essex County College, NJ
e-mail:
The speed of light according to special relativity has the same value with respect to any inertial frame of reference independent of whether it is associated with a distant star, the Earth or a source moving with respect to the Earth. Special relativity explains this paradox by proposing that our space is 4-dimensional pseudo-Euclidian (Minkowski space) and hence that the classical law of composition of velocities is not correct. In this work we attempt to build an alternative physical model in the framework of the model of the three-dimensional Euclidean space. We show mathematically that it is possible to derive the Fresnel formula on the basis of the classical mechanics law of composition of velocities. In addition we demonstrate on the basis of astronomical observations of binary stars and observations of the transverse Doppler Effect that the speed of light can change within a physical frame of reference. Also in this work we present arguments in favor of a model of an aether with properties similar to superfluid 3He.
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College Park, ND 2013 PROCEEDINGS of the NPA
1. Light still remains a “dark” issue in physics.
The speed of light according to special relativity (SR) has the same value with respect to any inertial frame of reference in Minkowski space. An attempt to build an alternative physical model in 3-dimensional Euclidean space brings us back to a classical problem: in what frame of reference does light travel with the speed? More than 100 years passed since the time this problem brought the so-called “crisis in physics” that was settled with the development of SR. During this time new ideas emerged and new experiments were performed among which there were some “problematic” experiments that contradicted the SR. However, it is conventional in the scientific community to consider a phenomenon as established provided it has been confirmed by several well known independent laboratories. For various reasons precisely these “problematic” experiments were not repeated in these laboratories.
From the model of the three-dimensional Euclidean space and independent time it follows that the speed of light in various geometric frames of reference may have any value. However, this conclusion requires a more detailed discussion when the real physical frames of reference are considered wherein experiments are conducted, specifically, those that demonstrate the invariance of the speed of light. Obviously, if propagation of light is some process in a medium (aether) then the motion of the aether itself with respect to a given frame of reference should be taken into consideration too. If propagation of light is a nonlinear process in a medium like a soliton (note that the density inside a soliton can be different than that of the surrounding medium) then the study of the affect of the aether’s moton on the speed of light becomes a very complicated problem.
It has been established that light transfers energy from one physical body, the source, to another, the receiver, in discrete increments, that is, quanta. However, among physicists there is no unified point of view for the description of the material carrier of the quantum, that is, the photon. There are three types of photon that are usually used in descriptions of the optical experiments demonstrating quantum properties of light [1]. The difference in usage of the term "photon" reflects the difference in interpretation of the results of such experiments.
1) The C-photon is a classical wave packet, that is, spatially localized, quasi monochromatic electromagnetic radiation carrying a quantum of energy, where is a central frequency of the radiation spectrum. The “corpuscular” properties of the C-photon reveal themselves only at the moment of detection.
But there are quantum optical effects: the essential quantum effects that have no classical analogues. Such effects cannot be described in the framework of the semi classical model based on Maxwell's equations
2) The M-photon is a hypothetical elementary particle of the light field generating an impulse at the output of the photodetector. Although there is no more rigorous definition of the M-photon in the framework of any consistent theory, the photon as a particle (with the wave properties characteristic of the elementary particles) is used in various optical studies where an attempt is made to go beyond the framework of the Copenhagen interpretation. Here, it is assumed that any radiation field consists of a set of almost independent M-photons with definite a priori features to be revealed after a time.
It is interesting that the first corpuscular models of the light field consisting of the elementary particles, each with energy where is the radiation frequency, were developed after A. Compton’s experiments on X-ray scattering (1922). The observed change in the frequency of the scattered radiation was explained by the elastic collision of an electron and a particle possessing energy and momentum. In 1929, G.H. Lewis called this particle a photon.
3) Q-photon is an objective entity corresponding to the Fock state of the light field with or a superposition of such states with nearly equal energies. This definition can be made in terms of the standard quantum theory of light. However, the statement that “light consists of photons” suggesting the definite number of such constituent elements of light does not make any sense in the standard quantum theory because the field has no definite before measurement. Of course, a problem of interpreting the quantum formalism still remains. The Copenhagen interpretation, accepted by most of the physicists, forbids asking nature “idle” questions, that is, it has a pragmatic tint. In the framework of this interpretation “A photon can be called a photon if only it is a detected photon”. Only investigating the characteristics of the pure or combined state of the field is permitted.
There is a case where all the above mentioned types of photon appear consistent: when the light field is in the one-photon state (photon in the pure state). In this case a priori properties of the photon can be discussed.
2. Hidden Dynamics in Relativistic Kinematics
In 1908 Walter von Ritz suggested that the speed was the speed of light with respect to the source and the classical law of composition of velocities is valid for the case of the moving source (the so called Ritz ballistic hypothesis) [2]. Under this assumption the aberration of starlight, the results of the famous Michelson-Morley experiment, and those of most other experiments aimed at detecting the aether wind come into agreement with each other.
However, the experiment performed at CERN, Geneva, in 1964 was considered to be the most convincing evidence against the Ritz theory. It follows from that experiment that the velocity of the photon equals c as measured with respect to the Earth. In this experiment the speed of 6 GeV photons produced in the decay of very energetic neutral pions was measured by time-of-flight over paths up to 80 meters in length. The pions were produced by the bombardment of a beryllium target with 19.2 GeV protons having speeds (inferred from the measured speeds of charged pions produced in the same bombardment) of 0.99975… [3]. Within experimental error it was found that the speed of the photons emitted by the extremely rapidly moving source was equal to. If the observed speed is written as , where is the speed of the source, the experiment showed
(1)
From the standpoint of SR the speed of аn emitted photon measured with respect to an inertial frame of reference associated with a source is always equal to , which agrees with the Ritz hypothesis. On the other hand, from SR it also follows that the speed of а photon measured with respect to Earth is also equal to , which agrees with the experiment performed at CERN. SR provides an explanation of the above two statements by discarding the classical law of composition of velocities and the hypothesis of aether as a preferred reference system, and introducing a model of four-dimensional pseudo-Euclidian space.
One of the main arguments in support of SR is the formula for the Doppler Effect. Indeed the formula for the transverse and longitudinal Doppler Effect
(2)
immediately follows from relativistic kinematics if light is considered as a wave process.
However a different explanation can be proposed. Relativistic kinematics is not correct, and it is possible to derive the transverse Doppler Effect remaining in the framework of the model of the three-dimensional Euclidean space and the classical law of composition of velocities. But in this case we have to assume that the speed of light (photon) can change within the same real physical frames of reference (for example leaving a source with a speed , where is a speed of the source, converges to the value near the Earth’s surface as is observed in experiments ). Generalizing the above it can be concluded from the fact that relativistic kinematics correctly describes the results of certain optical experiments that in the four-dimensional kinematic formalism of special relativity there are dynamics ‘hidden’ in the geometry of space. This idea was first put forward by E.L. Fainberg in 1997 [4]. Below we demonstrate it on the examples of the transverse Doppler Effect, and the astronomical observations of the binary stars motion.
3. Тhe derivation of the formula for the transverse and longitudinal Doppler Effect using the classical mechanics law of composition of velocities
Below, the equation for the transverse and longitudinal Doppler Effect is derived for the case of a photon in the pure state. In this case the properties of the photon can be discussed: its energy, momentum, mass, polarization. We assume that the classical law of composition of velocities and the law of conservation of energy and momentum are valid.
Case 1: Suppose that a source of light is at rest with respect to the Earth, and an observer is moving with a constant speed relative to the Earth. In the frame of reference of the Earth, the speed of the photon emitted by the source is equal to and there is no reason why it should change in the observer’s frame of reference prior to interaction of the photon and the detector.
We will work in the frame of reference of the observer. In the observer’s frame of reference a source of light with mass M is moving with velocity (Fig. 1). The energy of the source is composed of kinetic energy and internal energy of the excited atoms. Denote by the internal energy of the source after the photon is emitted. In addition the source undergoes recoil due to emission: its speed gains an increment of (where is the speed of the source after emission of the photon). From the laws of conservation of energy and momentum for the photon and the source respectively, it follows that
(3) (4)
where is the mass carried away by the photon emitted with speed с with respect to the source, is the photon energy in the observer’s frame of reference, and is the photon velocity in the same frame. Note that the vector is directed towards the observer.
Fig. 1
From Eq. (4) we obtain for :
(5)
After emission of the photon, the internal energy of the atom is decreased by the amount , where is the natural frequency of the atom, that is. Taking this along with Eq. (5) into account, Eq. (3) can be expressed as follows:
(6)
If the mass of the source is much greater than that of a photon, the terms containing may be ignored. In this approximation, Eq. (6) takes the form:
(7)
Using the relation (note that this is not a consequence of special relativity), Eq. (7) can be represented in two equivalent forms:
(8)
where (9)
Here is the angle between the velocity of the source and the direction from the source to the observer, i.e. the angle between vectors and.
Consider the special case . In this case Eq. (8) implies:
(10)
A very important result follows from Eq. (8) and Eq. (10): the energy of a photon, as an entity with mass, can be represented as a sum of two terms, the first being the kinetic energy of the center of mass, in which we assume all of the photon’s mass is concentrated; the second being the energy associated with the motion about the centre of mass, which is characteristic of the photon’s intrinsic degrees of freedom. Formula (10) was obtained by L.Boldyreva and N. Sotina in 1999. [5].
It is experimentally established that the absorption of light occurs in a quanta of energy , where is the detected frequency. Assume that all the energy of the photon is equal to the energy detected by a measuring device (this assumption is no different than that of conventional physics). Under this assumption, from equation (7) we obtain
(11)
If , that is, , then the expression for the transverse Doppler effect follows from Eq. (11):